If it happened after you updated a page, I would look at what changed with that page update. Did you add a widget or a shortcode or a plugin that’s displaying something on that page that doesn’t exist on other pages? Is the dropdown broken everywhere or just that page?

If I had to take a guess, I’d say there might be a javascript conflict or error occurring. I just checked your site now and the dropdowns appear to be working, so I can’t reproduce the problem unless it’s happening on a specific page. If that’s the case, then you should look at what’s on that page that could be causing a conflict.